Nonlinear Diffusion Equations: Full characterization of Entropies

Abstract

This paper is concerned with the large-time behavior of quasilinear Fokker-Planck equations with confinement on the whole space $\mathbb{R}^d$. It aims at characterizing all relative entropy functionals such that the entropy method à la Bakry-Émery yields exponential convergence of all solutions towards the unique steady state (with the same mass as the initial condition). We call such entropies admissible. The convergence rate is determined by the uniform convexity parameter of the confinement potential. As such, this program extends the analogous study of linear Fokker-Planck equations [Bakry-Émery, Arnold-Markowich-Toscani-Unterreiter] to the nonlinear case, and it derives additional functionals for the nonlinear case --- beyond the Ralston-Newman entropies used in [Jüngel-Carrillo-Markowich-Toscani-Unterreiter]. Two key results are the characterization of those nonlinear Fokker-Planck equations which admit all entropy functionals that are admissible for the corresponding linear Fokker-Planck equation, and vice versa, the characterization of all admissible entropies for a given nonlinearity. The latter quest for power-law nonlinearities yields a large family of entropies for the porous-medium equations, but only the Ralston-Newman entropy for the fast-diffusion equations. Additional results include the derivation of new generalized Csiszár-Kullback and generalized Log-Sobolev inequalities for our entropy functionals as well as moment-weighted $L^1$--convergence estimates for the Fokker-Planck solutions.